Your data matches 116 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000543
(load all 99 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => 1 = 0 + 1
[[1],[2]]
 => 1 => 1 = 0 + 1
[[1,2,3]]
 => 00 => 1 = 0 + 1
[[1,3],[2]]
 => 10 => 2 = 1 + 1
[[1,2],[3]]
 => 01 => 2 = 1 + 1
[[1],[2],[3]]
 => 11 => 1 = 0 + 1
[[1,2,3,4]]
 => 000 => 1 = 0 + 1
[[1,3,4],[2]]
 => 100 => 3 = 2 + 1
[[1,2,3],[4]]
 => 001 => 3 = 2 + 1
[[1,2],[3,4]]
 => 010 => 3 = 2 + 1
[[1,4],[2],[3]]
 => 110 => 3 = 2 + 1
[[1,2],[3],[4]]
 => 011 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => 111 => 1 = 0 + 1
[[1,2,3,4,5]]
 => 0000 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => 1111 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => 00000 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => 11111 => 1 = 0 + 1
Description
The size of the conjugacy class of a binary word. Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
Matching statistic: St001958
(load all 160 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => 1 = 0 + 1
[[1],[2]]
 => [2,1] => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 1 = 0 + 1
Description
The degree of the polynomial interpolating the values of a permutation. Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$. This statistic records the degree of $p$.
Matching statistic: St000293
(load all 3 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => 0 => 0
[[1],[2]]
 => 1 => 1 => 0
[[1,2,3]]
 => 00 => 00 => 0
[[1,3],[2]]
 => 10 => 10 => 1
[[1,2],[3]]
 => 01 => 10 => 1
[[1],[2],[3]]
 => 11 => 11 => 0
[[1,2,3,4]]
 => 000 => 000 => 0
[[1,3,4],[2]]
 => 100 => 100 => 2
[[1,2,3],[4]]
 => 001 => 100 => 2
[[1,2],[3,4]]
 => 010 => 100 => 2
[[1,4],[2],[3]]
 => 110 => 110 => 2
[[1,2],[3],[4]]
 => 011 => 110 => 2
[[1],[2],[3],[4]]
 => 111 => 111 => 0
[[1,2,3,4,5]]
 => 0000 => 0000 => 0
[[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 0
[[1,2,3,4,5,6]]
 => 00000 => 00000 => 0
[[1],[2],[3],[4],[5],[6]]
 => 11111 => 11111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St000431
(load all 6 compositions to match this statistic)
Mapping
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000431: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => {{1,2}}
 => [2,1] => 0
[[1],[2]]
 => {{1},{2}}
 => [1,2] => 0
[[1,2,3]]
 => {{1,2,3}}
 => [2,3,1] => 0
[[1,3],[2]]
 => {{1,3},{2}}
 => [3,2,1] => 1
[[1,2],[3]]
 => {{1,2},{3}}
 => [2,1,3] => 1
[[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => 0
[[1,2,3,4]]
 => {{1,2,3,4}}
 => [2,3,4,1] => 0
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => [3,2,4,1] => 2
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => [2,3,1,4] => 2
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 2
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 0
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => 0
[[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => 0
Description
The number of occurrences of the pattern 213 or of the pattern 321 in a permutation.
Matching statistic: St000682
(load all 4 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => 0 => 0
[[1],[2]]
 => 1 => 1 => 0
[[1,2,3]]
 => 00 => 00 => 0
[[1,3],[2]]
 => 10 => 01 => 1
[[1,2],[3]]
 => 01 => 01 => 1
[[1],[2],[3]]
 => 11 => 11 => 0
[[1,2,3,4]]
 => 000 => 000 => 0
[[1,3,4],[2]]
 => 100 => 001 => 2
[[1,2,3],[4]]
 => 001 => 001 => 2
[[1,2],[3,4]]
 => 010 => 001 => 2
[[1,4],[2],[3]]
 => 110 => 011 => 2
[[1,2],[3],[4]]
 => 011 => 011 => 2
[[1],[2],[3],[4]]
 => 111 => 111 => 0
[[1,2,3,4,5]]
 => 0000 => 0000 => 0
[[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 0
[[1,2,3,4,5,6]]
 => 00000 => 00000 => 0
[[1],[2],[3],[4],[5],[6]]
 => 11111 => 11111 => 0
Description
The Grundy value of Welter's game on a binary word. Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
Matching statistic: St000848
(load all 3 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000848: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => ([(0,1)],2)
 => 0
[[1],[2]]
 => 1 => ([(0,1)],2)
 => 0
[[1,2,3]]
 => 00 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2]]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,2],[3]]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1],[2],[3]]
 => 11 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4]]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,4],[2]]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2,3],[4]]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3,4]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[[1,4],[2],[3]]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3],[4]]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1],[2],[3],[4]]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,3,4,5]]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1],[2],[3],[4],[5]]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,2,3,4,5,6]]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
The balance constant multiplied with the number of linear extensions of a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$ Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced. Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
(load all 5 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => ([(0,1)],2)
 => 0
[[1],[2]]
 => 1 => ([(0,1)],2)
 => 0
[[1,2,3]]
 => 00 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2]]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,2],[3]]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1],[2],[3]]
 => 11 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4]]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,4],[2]]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2,3],[4]]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3,4]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[[1,4],[2],[3]]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3],[4]]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1],[2],[3],[4]]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,3,4,5]]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1],[2],[3],[4],[5]]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,2,3,4,5,6]]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
The number of 1/3-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001436
(load all 3 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St001436: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => 0 => 0
[[1],[2]]
 => 1 => 1 => 0
[[1,2,3]]
 => 00 => 00 => 0
[[1,3],[2]]
 => 10 => 10 => 1
[[1,2],[3]]
 => 01 => 10 => 1
[[1],[2],[3]]
 => 11 => 11 => 0
[[1,2,3,4]]
 => 000 => 000 => 0
[[1,3,4],[2]]
 => 100 => 100 => 2
[[1,2,3],[4]]
 => 001 => 100 => 2
[[1,2],[3,4]]
 => 010 => 100 => 2
[[1,4],[2],[3]]
 => 110 => 110 => 2
[[1,2],[3],[4]]
 => 011 => 110 => 2
[[1],[2],[3],[4]]
 => 111 => 111 => 0
[[1,2,3,4,5]]
 => 0000 => 0000 => 0
[[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 0
[[1,2,3,4,5,6]]
 => 00000 => 00000 => 0
[[1],[2],[3],[4],[5],[6]]
 => 11111 => 11111 => 0
Description
The index of a given binary word in the lex-order among all its cyclic shifts.
Matching statistic: St001633
(load all 4 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 0 => ([(0,1)],2)
 => 0
[[1],[2]]
 => 1 => ([(0,1)],2)
 => 0
[[1,2,3]]
 => 00 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2]]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,2],[3]]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1],[2],[3]]
 => 11 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4]]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,4],[2]]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2,3],[4]]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3,4]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[[1,4],[2],[3]]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,2],[3],[4]]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1],[2],[3],[4]]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,3,4,5]]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1],[2],[3],[4],[5]]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,2,3,4,5,6]]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1],[2],[3],[4],[5],[6]]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000003
(load all 3 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3]
 => 1 = 0 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => 3 = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4]
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5]
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 1 = 0 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6]
 => 1 = 0 + 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000626The minimal period of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001313The number of Dyck paths above the lattice path given by a binary word. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000057The Shynar inversion number of a standard tableau. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000290The major index of a binary word. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001377The major index minus the number of inversions of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001485The modular major index of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001742The difference of the maximal and the minimal degree in a graph. St001910The height of the middle non-run of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000100The number of linear extensions of a poset. St000255The number of reduced Kogan faces with the permutation as type. St000277The number of ribbon shaped standard tableaux. St000525The number of posets with the same zeta polynomial. St000548The number of different non-empty partial sums of an integer partition. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001268The size of the largest ordinal summand in the poset. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St001595The number of standard Young tableaux of the skew partition. St001779The order of promotion on the set of linear extensions of a poset. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001760The number of prefix or suffix reversals needed to sort a permutation. St000045The number of linear extensions of a binary tree. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001535The number of cyclic alignments of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001623The number of doubly irreducible elements of a lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree.